{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,8,26]],"date-time":"2025-08-26T06:42:09Z","timestamp":1756190529324,"version":"3.37.3"},"reference-count":62,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,2,16]],"date-time":"2022-02-16T00:00:00Z","timestamp":1644969600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,2,16]],"date-time":"2022-02-16T00:00:00Z","timestamp":1644969600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Stat Comput"],"published-print":{"date-parts":[[2022,4,15]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Spectral embedding of network adjacency matrices often produces node representations living approximately around low-dimensional submanifold structures. In particular, hidden substructure is expected to arise when the graph is generated from a latent position model. Furthermore, the presence of communities within the network might generate community-specific submanifold structures in the embedding, but this is not explicitly accounted for in most statistical models for networks. In this article, a class of models called latent structure block models (LSBM) is proposed to address such scenarios, allowing for graph clustering when community-specific one-dimensional manifold structure is present. LSBMs focus on a specific class of latent space model, the random dot product graph (RDPG), and assign a latent submanifold to the latent positions of each community. A Bayesian model for the embeddings arising from LSBMs is discussed, and shown to have a good performance on simulated and real-world network data. The model is able to correctly recover the underlying communities living in a one-dimensional manifold, even when the parametric form of the underlying curves is unknown, achieving remarkable results on a variety of real data.\n<\/jats:p>","DOI":"10.1007\/s11222-022-10082-6","type":"journal-article","created":{"date-parts":[[2022,3,15]],"date-time":"2022-03-15T16:10:46Z","timestamp":1647360646000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Latent structure blockmodels for Bayesian spectral graph clustering"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4571-6681","authenticated-orcid":false,"given":"Francesco","family":"Sanna Passino","sequence":"first","affiliation":[]},{"given":"Nicholas A.","family":"Heard","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2022,2,16]]},"reference":[{"issue":"1","key":"10082_CR1","doi-asserted-by":"publisher","first-page":"531","DOI":"10.1214\/20-AOS1967","volume":"49","author":"AA Amini","year":"2021","unstructured":"Amini, A.A., Razaee, Z.S.: Concentration of kernel matrices with application to kernel spectral clustering. Ann. Stat. 49(1), 531\u2013556 (2021)","journal-title":"Ann. Stat."},{"key":"10082_CR2","unstructured":"Asta, D.M., Shalizi, C.R.: Geometric network comparisons. In: Proceedings of the Thirty-First Conference on Uncertainty in Artificial Intelligence. pp. 102\u2013110. UAI\u201915, AUAI Press (2015)"},{"issue":"1","key":"10082_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s13171-015-0071-x","volume":"78","author":"A Athreya","year":"2016","unstructured":"Athreya, A., Priebe, C.E., Tang, M., Lyzinski, V., Marchette, D.J., Sussman, D.L.: A limit theorem for scaled eigenvectors of random dot product graphs. Sankhya A 78(1), 1\u201318 (2016)","journal-title":"Sankhya A"},{"issue":"226","key":"10082_CR4","first-page":"1","volume":"18","author":"A Athreya","year":"2018","unstructured":"Athreya, A., Fishkind, D.E., Tang, M., Priebe, C.E., Park, Y., Vogelstein, J.T., Levin, K., Lyzinski, V., Qin, Y., Sussman, D.L.: Statistical inference on random dot product graphs: a survey. J. Mach. Learn. Res. 18(226), 1\u201392 (2018)","journal-title":"J. Mach. Learn. Res."},{"issue":"1","key":"10082_CR5","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1214\/20-STS787","volume":"36","author":"A Athreya","year":"2021","unstructured":"Athreya, A., Tang, M., Park, Y., Priebe, C.E.: On estimation and inference in latent structure random graphs. Stat. Sci. 36(1), 68\u201388 (2021)","journal-title":"Stat. Sci."},{"key":"10082_CR6","doi-asserted-by":"publisher","first-page":"P10008","DOI":"10.1088\/1742-5468\/2008\/10\/P10008","volume":"10","author":"VD Blondel","year":"2008","unstructured":"Blondel, V.D., Guillaume, J.L., Lambiotte, R., Lefebvre, E.: Fast unfolding of communities in large networks. J. Stat. Mech. Theory Exp. 10, P10008 (2008)","journal-title":"J. Stat. Mech. Theory Exp."},{"issue":"6","key":"10082_CR7","doi-asserted-by":"publisher","first-page":"1143","DOI":"10.1007\/s11222-014-9505-x","volume":"25","author":"C Bouveyron","year":"2015","unstructured":"Bouveyron, C., Fauvel, M., Girard, S.: Kernel discriminant analysis and clustering with parsimonious Gaussian process models. Stat. Comput. 25(6), 1143\u20131162 (2015)","journal-title":"Stat. Comput."},{"key":"10082_CR8","unstructured":"Cheng, C.A., Boots, B.: Variational inference for gaussian process models with linear complexity. In: Proceedings of the 31st International Conference on Neural Information Processing Systems. pp. 5190\u20135200. NIPS\u201917, Curran Associates Inc., Red Hook, NY, USA (2017)"},{"issue":"1","key":"10082_CR9","doi-asserted-by":"publisher","first-page":"1393","DOI":"10.1214\/16-EJS1144","volume":"10","author":"R Couillet","year":"2016","unstructured":"Couillet, R., Benaych-Georges, F.: Kernel spectral clustering of large dimensional data. Electron. J. Stat. 10(1), 1393\u20131454 (2016)","journal-title":"Electron. J. Stat."},{"key":"10082_CR10","doi-asserted-by":"publisher","first-page":"1085","DOI":"10.2307\/2533007","volume":"51","author":"P Dellaportas","year":"1995","unstructured":"Dellaportas, P., Stephens, D.: Bayesian analysis of errors-in-variables regression models. Biometrics 51, 1085\u20131095 (1995)","journal-title":"Biometrics"},{"issue":"2","key":"10082_CR11","first-page":"19","volume":"19","author":"E Diday","year":"1971","unstructured":"Diday, E.: Une nouvelle m\u00e9thode en classification automatique et reconnaissance des formes la m\u00e9thode des nu\u00e9es dynamiques. Revue de Statistique Appliqu\u00e9e 19(2), 19\u201333 (1971)","journal-title":"Revue de Statistique Appliqu\u00e9e"},{"key":"10082_CR12","first-page":"47","volume-title":"Clustering analysis","author":"E Diday","year":"1976","unstructured":"Diday, E., Simon, J.C.: Clustering analysis, pp. 47\u201394. Springer, Berlin Heidelberg, Berlin, Heidelberg (1976)"},{"key":"10082_CR13","doi-asserted-by":"crossref","unstructured":"Dryden, I.L., Mardia, K.V.: Statistical shape analysis, with applications in R. John Wiley and Sons (2016)","DOI":"10.1002\/9781119072492"},{"key":"10082_CR14","unstructured":"Dugu\u00e9, N., Perez, A.: Directed Louvain: maximizing modularity in directed networks. Tech. Rep. hal-01231784, Universit\u00e9 d\u2019Orl\u00e9ans (2015)"},{"key":"10082_CR15","unstructured":"Dunson, D.B., Wu, N.: Inferring manifolds from noisy data using Gaussian processes. arXiv e-prints (2021)"},{"issue":"7666","key":"10082_CR16","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1038\/nature23455","volume":"548","author":"K Eichler","year":"2017","unstructured":"Eichler, K., Li, F., Litwin-Kumar, A., Park, Y., Andrade, I., Schneider-Mizell, C.M., Saumweber, T., Huser, A., Eschbach, C., Gerber, B., Fetter, R.D., Truman, J.W., Priebe, C.E., Abbott, L.F., Thum, A.S., Zlatic, M., Cardona, A.: The complete connectome of a learning and memory centre in an insect brain. Nature 548(7666), 175\u2013182 (2017)","journal-title":"Nature"},{"issue":"0","key":"10082_CR17","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1080\/01621459.2021.1917418","volume":"0","author":"X Han","year":"2021","unstructured":"Han, X., Tong, X., Fan, Y.: Eigen selection in spectral clustering: a theory-guided practice. J. Am. Stat. Assoc. 0(0), 1\u201313 (2021)","journal-title":"J. Am. Stat. Assoc."},{"issue":"460","key":"10082_CR18","doi-asserted-by":"publisher","first-page":"1090","DOI":"10.1198\/016214502388618906","volume":"97","author":"PD Hoff","year":"2002","unstructured":"Hoff, P.D., Raftery, A.E., Handcock, M.S.: Latent space approaches to social network analysis. J. Am. Stat. Assoc. 97(460), 1090\u20131098 (2002)","journal-title":"J. Am. Stat. Assoc."},{"issue":"4","key":"10082_CR19","doi-asserted-by":"publisher","first-page":"980","DOI":"10.1080\/10618600.2019.1593180","volume":"28","author":"DP Hofmeyr","year":"2019","unstructured":"Hofmeyr, D.P.: Improving spectral clustering using the asymptotic value of the normalized cut. J. Comput. Graph. Stat. 28(4), 980\u2013992 (2019)","journal-title":"J. Comput. Graph. Stat."},{"issue":"2","key":"10082_CR20","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/0378-8733(83)90021-7","volume":"5","author":"PW Holland","year":"1983","unstructured":"Holland, P.W., Laskey, K.B., Leinhardt, S.: Stochastic blockmodels: first steps. Soc. Netw. 5(2), 109\u2013137 (1983)","journal-title":"Soc. Netw."},{"issue":"1","key":"10082_CR21","doi-asserted-by":"publisher","first-page":"193","DOI":"10.1007\/BF01908075","volume":"2","author":"L Hubert","year":"1985","unstructured":"Hubert, L., Arabie, P.: Comparing partitions. J. Classif. 2(1), 193\u2013218 (1985)","journal-title":"J. Classif."},{"issue":"1","key":"10082_CR22","doi-asserted-by":"publisher","first-page":"50","DOI":"10.1214\/088342305000000016","volume":"20","author":"A Jasra","year":"2005","unstructured":"Jasra, A., Holmes, C.C., Stephens, D.A.: Markov Chain Monte Carlo methods and the label switching problem in Bayesian mixture modeling. Stat. Sci. 20(1), 50\u201367 (2005)","journal-title":"Stat. Sci."},{"key":"10082_CR23","unstructured":"Jones, A., Rubin-Delanchy, P.: The multilayer random dot product graph. arXiv e-prints (2020)"},{"issue":"1","key":"10082_CR24","doi-asserted-by":"publisher","first-page":"016107","DOI":"10.1103\/PhysRevE.83.016107","volume":"83","author":"B Karrer","year":"2011","unstructured":"Karrer, B., Newman, M.E.J.: Stochastic blockmodels and community structure in networks. Phys. Rev. E 83(1), 016107 (2011)","journal-title":"Phys. Rev. E"},{"issue":"6","key":"10082_CR25","doi-asserted-by":"publisher","first-page":"1201","DOI":"10.1007\/s11222-014-9472-2","volume":"25","author":"C Keribin","year":"2015","unstructured":"Keribin, C., Brault, V., Celeux, G., Govaert, G.: Estimation and selection for the latent block model on categorical data. Stat. Comput. 25(6), 1201\u20131216 (2015)","journal-title":"Stat. Comput."},{"issue":"4","key":"10082_CR26","doi-asserted-by":"publisher","first-page":"887","DOI":"10.1214\/aoms\/1177728066","volume":"27","author":"J Kiefer","year":"1956","unstructured":"Kiefer, J., Wolfowitz, J.: Consistency of the maximum likelihood estimator in the presence of infinitely many incidental parameters. Ann. Math. Stat. 27(4), 887\u2013906 (1956)","journal-title":"Ann. Math. Stat."},{"issue":"1","key":"10082_CR27","doi-asserted-by":"publisher","first-page":"93","DOI":"10.1177\/1471082X1001200105","volume":"12","author":"P Latouche","year":"2012","unstructured":"Latouche, P., Birmel\u00e9, E., Ambroise, C.: Variational Bayesian inference and complexity control for stochastic block models. Stat. Model. 12(1), 93\u2013115 (2012)","journal-title":"Stat. Model."},{"issue":"4","key":"10082_CR28","doi-asserted-by":"publisher","first-page":"1386","DOI":"10.1016\/j.patcog.2011.10.004","volume":"45","author":"M L\u00e1zaro-Gredilla","year":"2012","unstructured":"L\u00e1zaro-Gredilla, M., Van Vaerenbergh, S., Lawrence, N.D.: Overlapping mixtures of Gaussian processes for the data association problem. Pattern Recogn. 45(4), 1386\u20131395 (2012)","journal-title":"Pattern Recogn."},{"issue":"1","key":"10082_CR29","doi-asserted-by":"publisher","first-page":"86","DOI":"10.1214\/aos\/1176346059","volume":"11","author":"BG Lindsay","year":"1983","unstructured":"Lindsay, B.G.: The geometry of mixture likelihoods: a general theory. Ann. Stat. 11(1), 86\u201394 (1983)","journal-title":"Ann. Stat."},{"issue":"10","key":"10082_CR30","doi-asserted-by":"publisher","first-page":"2213","DOI":"10.1016\/S0031-3203(03)00084-0","volume":"36","author":"B Luo","year":"2003","unstructured":"Luo, B., Wilson, R.C., Hancock, E.R.: Spectral embedding of graphs. Pattern Recognit. 36(10), 2213\u20132230 (2003)","journal-title":"Pattern Recognit."},{"issue":"2","key":"10082_CR31","doi-asserted-by":"publisher","first-page":"2905","DOI":"10.1214\/14-EJS978","volume":"8","author":"V Lyzinski","year":"2014","unstructured":"Lyzinski, V., Sussman, D.L., Tang, M., Athreya, A., Priebe, C.E.: Perfect clustering for stochastic blockmodel graphs via adjacency spectral embedding. Electron. J. Stat. 8(2), 2905\u20132922 (2014)","journal-title":"Electron. J. Stat."},{"issue":"512","key":"10082_CR32","doi-asserted-by":"publisher","first-page":"1684","DOI":"10.1080\/01621459.2014.991395","volume":"110","author":"TH McCormick","year":"2015","unstructured":"McCormick, T.H., Zheng, T.: Latent surface models for networks using aggregated relational data. J. Am. Stat. Assoc. 110(512), 1684\u20131695 (2015)","journal-title":"J. Am. Stat. Assoc."},{"issue":"8","key":"10082_CR33","doi-asserted-by":"publisher","first-page":"1222","DOI":"10.1093\/bioinformatics\/bth068","volume":"20","author":"M Medvedovic","year":"2004","unstructured":"Medvedovic, M., Yeung, K.Y., Bumgarner, R.E.: Bayesian mixture model based clustering of replicated microarray data. Bioinformatics 20(8), 1222\u20131232 (2004)","journal-title":"Bioinformatics"},{"key":"10082_CR34","unstructured":"Modell, A., Rubin-Delanchy, P.: Spectral clustering under degree heterogeneity: a case for the random walk Laplacian. arXiv e-prints (2021)"},{"key":"10082_CR35","unstructured":"Ng, A.Y., Jordan, M.I., Weiss, Y.: On spectral clustering: Analysis and an algorithm. In: Proceedings of the 14th International Conference on Neural Information Processing Systems. pp. 849\u2013856 (2001)"},{"issue":"1","key":"10082_CR36","doi-asserted-by":"publisher","first-page":"678","DOI":"10.1214\/19-EJS1533","volume":"13","author":"M Pensky","year":"2019","unstructured":"Pensky, M., Zhang, T.: Spectral clustering in the dynamic stochastic block model. Electron. J. Stat. 13(1), 678\u2013709 (2019)","journal-title":"Electron. J. Stat."},{"key":"10082_CR37","unstructured":"Priebe, C.E., Park, Y., Tang, M., Athreya, A., Lyzinski, V., Vogelstein, J.T., Qin, Y., Cocanougher, B., Eichler, K., Zlatic, M., Cardona, A.: Semiparametric spectral modeling of the Drosophila connectome. arXiv e-prints (2017)"},{"issue":"13","key":"10082_CR38","doi-asserted-by":"publisher","first-page":"5995","DOI":"10.1073\/pnas.1814462116","volume":"116","author":"CE Priebe","year":"2019","unstructured":"Priebe, C.E., Park, Y., Vogelstein, J.T., Conroy, J.M., Lyzinski, V., Tang, M., Athreya, A., Cape, J., Bridgeford, E.: On a two-truths phenomenon in spectral graph clustering. Proc. Natl. Acad. Sci. 116(13), 5995\u20136000 (2019)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"10082_CR39","unstructured":"Qin, T., Rohe, K.: Regularized spectral clustering under the degree-corrected stochastic blockmodel. In: Proceedings of the 26th International Conference on Neural Information Processing Systems. vol.\u00a02, pp. 3120\u20133128 (2013)"},{"key":"10082_CR40","doi-asserted-by":"crossref","unstructured":"Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Adaptive Computation and Machine Learning (2006)","DOI":"10.7551\/mitpress\/3206.001.0001"},{"issue":"4","key":"10082_CR41","doi-asserted-by":"publisher","first-page":"1878","DOI":"10.1214\/11-AOS887","volume":"39","author":"K Rohe","year":"2011","unstructured":"Rohe, K., Chatterjee, S., Yu, B.: Spectral clustering and the high-dimensional stochastic blockmodel. Ann. Stat. 39(4), 1878\u20131915 (2011)","journal-title":"Ann. Stat."},{"key":"10082_CR42","unstructured":"Ross, J.C., Dy, J.G.: Nonparametric mixture of Gaussian processes with constraints. In: Proceedings of the 30th International Conference on Machine Learning - Volume 28. ICML\u201913 (2013)"},{"key":"10082_CR43","unstructured":"Rubin-Delanchy, P.: Manifold structure in graph embeddings. In: Larochelle, H., Ranzato, M., Hadsell, R., Balcan, M.F., Lin, H. (eds.) Advances in Neural Information Processing Systems. vol.\u00a033, pp. 11687\u201311699. Curran Associates, Inc. (2020)"},{"key":"10082_CR44","unstructured":"Rubin-Delanchy, P., Cape, J., Tang, M., Priebe, C.E.: A statistical interpretation of spectral embedding: the generalised random dot product graph. arXiv e-prints (2017)"},{"issue":"3","key":"10082_CR45","doi-asserted-by":"publisher","first-page":"1217","DOI":"10.1214\/16-AOAS955","volume":"11","author":"M Salter-Townshend","year":"2017","unstructured":"Salter-Townshend, M., McCormick, T.H.: Latent space models for multiview network data. Ann. Appl. Stat. 11(3), 1217\u20131244 (2017)","journal-title":"Ann. Appl. Stat."},{"issue":"5","key":"10082_CR46","doi-asserted-by":"publisher","first-page":"1291","DOI":"10.1007\/s11222-020-09946-6","volume":"30","author":"F Sanna Passino","year":"2020","unstructured":"Sanna Passino, F., Heard, N.A.: Bayesian estimation of the latent dimension and communities in stochastic blockmodels. Stat. Comput. 30(5), 1291\u20131307 (2020)","journal-title":"Stat. Comput."},{"key":"10082_CR47","doi-asserted-by":"crossref","unstructured":"Sanna Passino, F., Heard, N.A., Rubin-Delanchy, P.: Spectral clustering on spherical coordinates under the degree-corrected stochastic blockmodel. Technometrics (to appear) (2021)","DOI":"10.1080\/00401706.2021.2008503"},{"key":"10082_CR48","doi-asserted-by":"crossref","unstructured":"Scholkopf, B., Smola, A.J.: Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Adaptive Computation and Machine Learning series (2018)","DOI":"10.7551\/mitpress\/4175.001.0001"},{"key":"10082_CR49","doi-asserted-by":"crossref","unstructured":"Shepard, R.N.: The analysis of proximities: Multidimensional scaling with an unknown distance function. i. Psychometrika 27(2), 125\u2013140 (1962)","DOI":"10.1007\/BF02289630"},{"issue":"3","key":"10082_CR50","doi-asserted-by":"publisher","first-page":"428","DOI":"10.1214\/19-STS702","volume":"34","author":"AL Smith","year":"2019","unstructured":"Smith, A.L., Asta, D.M., Calder, C.A.: The geometry of continuous latent space models for network data. Stat. Sci. 34(3), 428\u2013453 (2019)","journal-title":"Stat. Sci."},{"issue":"499","key":"10082_CR51","doi-asserted-by":"publisher","first-page":"1119","DOI":"10.1080\/01621459.2012.699795","volume":"107","author":"DL Sussman","year":"2012","unstructured":"Sussman, D.L., Tang, M., Fishkind, D.E., Priebe, C.E.: A consistent adjacency spectral embedding for stochastic blockmodel graphs. J. Am. Stat. Assoc. 107(499), 1119\u20131128 (2012)","journal-title":"J. Am. Stat. Assoc."},{"issue":"5500","key":"10082_CR52","doi-asserted-by":"publisher","first-page":"2319","DOI":"10.1126\/science.290.5500.2319","volume":"290","author":"JB Tenenbaum","year":"2000","unstructured":"Tenenbaum, J.B., de Silva, V., Langford, J.C.: A global geometric framework for nonlinear dimensionality reduction. Science 290(5500), 2319\u20132323 (2000)","journal-title":"Science"},{"issue":"2","key":"10082_CR53","doi-asserted-by":"publisher","first-page":"487","DOI":"10.1111\/rssb.12363","volume":"82","author":"A Todeschini","year":"2020","unstructured":"Todeschini, A., Miscouridou, X., Caron, F.: Exchangeable random measures for sparse and modular graphs with overlapping communities. J. Roy. Stat. Soc. B 82(2), 487\u2013520 (2020)","journal-title":"J. Roy. Stat. Soc. B"},{"issue":"4","key":"10082_CR54","doi-asserted-by":"publisher","first-page":"401","DOI":"10.1007\/BF02288916","volume":"17","author":"WS Torgerson","year":"1952","unstructured":"Torgerson, W.S.: Multidimensional scaling: I. Theory and method. Psychometrika 17(4), 401\u2013419 (1952)","journal-title":"Psychometrika"},{"key":"10082_CR55","unstructured":"Trosset, M.W., Gao, M., Tang, M., Priebe, C.E.: Learning 1-dimensional submanifolds for subsequent inference on random dot product graphs. arXiv e-prints (2020)"},{"issue":"4","key":"10082_CR56","doi-asserted-by":"publisher","first-page":"395","DOI":"10.1007\/s11222-007-9033-z","volume":"1","author":"U von Luxburg","year":"2007","unstructured":"von Luxburg, U.: A tutorial on spectral clustering. Stat. Comput. 1(4), 395\u2013416 (2007)","journal-title":"Stat. Comput."},{"key":"10082_CR57","doi-asserted-by":"crossref","unstructured":"Wyse, J., Friel, N., Latouche, P.: Inferring structure in bipartite networks using the latent blockmodel and exact ICL. Netw. Sci. 5(1), 45\u201369 (2017)","DOI":"10.1017\/nws.2016.25"},{"issue":"2","key":"10082_CR58","doi-asserted-by":"publisher","first-page":"422","DOI":"10.1080\/10618600.2020.1824870","volume":"30","author":"C Yang","year":"2021","unstructured":"Yang, C., Priebe, C.E., Park, Y., Marchette, D.J.: Simultaneous dimensionality and complexity model selection for spectral graph clustering. J. Comput. Graph. Stat. 30(2), 422\u2013441 (2021)","journal-title":"J. Comput. Graph. Stat."},{"key":"10082_CR59","doi-asserted-by":"publisher","first-page":"8506","DOI":"10.1109\/TIP.2020.3016491","volume":"29","author":"X Ye","year":"2020","unstructured":"Ye, X., Zhao, J., Chen, Y., Guo, L.J.: Bayesian adversarial spectral clustering with unknown cluster number. IEEE Trans. Image Process. 29, 8506\u20138518 (2020)","journal-title":"IEEE Trans. Image Process."},{"key":"10082_CR60","doi-asserted-by":"publisher","first-page":"138","DOI":"10.1007\/978-3-540-77004-6_11","volume-title":"Algorithms and Models for the Web-Graph","author":"SJ Young","year":"2007","unstructured":"Young, S.J., Scheinerman, E.R.: Random dot product graph models for social networks. In: Bonato, A., Chung, F.R.K. (eds.) Algorithms and Models for the Web-Graph, pp. 138\u2013149. Springer, Berlin Heidelberg, Berlin, Heidelberg (2007)"},{"issue":"2","key":"10082_CR61","doi-asserted-by":"publisher","first-page":"918","DOI":"10.1016\/j.csda.2005.09.010","volume":"51","author":"M Zhu","year":"2006","unstructured":"Zhu, M., Ghodsi, A.: Automatic dimensionality selection from the scree plot via the use of profile likelihood. Comput. Stat. Data. Anal. 51(2), 918\u2013930 (2006)","journal-title":"Comput. Stat. Data. Anal."},{"issue":"8","key":"10082_CR62","doi-asserted-by":"publisher","first-page":"1532","DOI":"10.1109\/TKDE.2018.2858782","volume":"31","author":"X Zhu","year":"2019","unstructured":"Zhu, X., Zhang, S., Li, Y., Zhang, J., Yang, L., Fang, Y.: Low-rank sparse subspace for spectral clustering. IEEE Trans. Knowl. Data Eng. 31(8), 1532\u20131543 (2019)","journal-title":"IEEE Trans. Knowl. Data Eng."}],"container-title":["Statistics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10082-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11222-022-10082-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11222-022-10082-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,4,21]],"date-time":"2022-04-21T15:18:08Z","timestamp":1650554288000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11222-022-10082-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,16]]},"references-count":62,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,4,15]]}},"alternative-id":["10082"],"URL":"https:\/\/doi.org\/10.1007\/s11222-022-10082-6","relation":{},"ISSN":["0960-3174","1573-1375"],"issn-type":[{"type":"print","value":"0960-3174"},{"type":"electronic","value":"1573-1375"}],"subject":[],"published":{"date-parts":[[2022,2,16]]},"assertion":[{"value":"7 July 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"22"}}